Structural Bioinformatics Library
Template C++ / Python API for developping structural bioinformatics applications.

Authors: F. Cazals and T. O'Donnell
Centers of mass and their generalization.
The celebrated center of mass of a point set in a Euclidean space is the (a) point minimizing the sum of squared Euclidean to points in this set.
Similarly, the mean (resp. pmean) of a point set is a metric space is the point minimizing the sum of squared distances (resp. pth powers of distances) to a finite point set.
This package provides a method to compute these quantities for points located on the unit circle , following the algorithm from [38] .
Definitions. Consider angles . Also consider the embedding on an angle onto the unit circle, that is .
The geodesic distance between the two points and on , denoted , satisfies
This distance is also directly expressed using the angles:
For an integer , consider the function involving the pth powers of distances to all points, that is
Consider the minimum of this function, that is
The value obtained for , corresponds to the mean. For , the minimum is the socalled pmean.
Sum of square distances function Sample points are in red, their antipodal points on the circle are in black, minimas are in blue and the larger blue dot is the global minima. In blue , In green and in yellow . 
Sum of distances to the power 5 function Sample points are in red, their antipodal points on the circle are in black, minimas are in blue and the larger blue dot is the global minima. In blue , In green and in yellow . 
In short, the algorithm [38] subdivides the unit circle into interval namely circle arcs defined by data points and their antipodal values. This set of angles is denoted . These intervals are such that there is at most one local minimum of on each interval. To spot these local minima efficiently, the algorithm maintains a polynomial expression of the function and its derivative . In short, the steps are:
Albeit simple, the algorithm is numerically challenging since it deals with piecewise polynomials with transcendental coefficients (involving ).
Following the classical design pattern in computational geometry, this package provides a generic implementation, templated by a kernel providing number types and associated predicates and constructions.
The main class is SBL::GT::T_Frechet_mean_S1.
Two kernels are provided:
Note that in our case, an erroneous evaluation of predicates may yield the loss or erroneous identification of a local minimum.
This package provides two executables:
This corresponds to the two above mentionned kernels.
See the following jupyter notebook:
import re #regular expressions
import sys #misc system
import os
import shutil
#import shutil # python 3 only
#input filename and output directory
odir = "results"
if not os.path.exists(odir):
os.system("mkdir results")
#choose value of p
p=4
angles_file="data/angles.txt" #file containing angles
weights_file="data/weights.txt" #file containing weights
exe_file="sblFrechetmeanusingrealvalueNT.exe" #App to compute minimas (sblFrechetmeanusingintervalNT.exe is another option)
# run command(r is used for specifiying radii input data a to get all minimas)
cmd = "./%s f %s w %s r p %d a o %s" % (exe_file,angles_file,weights_file,p,odir)
os.system(cmd)
# load visualisation script and instanciate using data and corresponding minimas
minimas_file="results/angles_local_minima.txt"
load("centroidcircle.sage")
algo = Project()
algo.load_angles(angles_file,False)
algo.load_weights(weights_file)
algo.load_minimas(minimas_file,False)
algo.run(p)
#change value of p
p=5
# rerun command(r is used for specifiying radii input data)
cmd = "./%s f %s w %s r p %d a o %s" % (exe_file,angles_file,weights_file,p,odir)
algo.run(p)